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In reply to "Re: Re: Re: Re: Re: rings, ideals and correspondence theorem -- clarification requested ", posted by gp on August 5, 2008:
>In reply to "Re: Re: Re: Re: rings, ideals and correspondence theorem -- clarification requested ", posted by Bill Dubuque on August 5, 2008:
>>In reply to "Re: Re: Re: rings, ideals and correspondence theorem -- clarification requested ", posted by Tonio on August 5, 2008:
>>>it is a standard theorem, though not a simple or trivial one, that any RNG (what most
>>>authors call "ring") can be embedded in a ring with unity, so it's really not that
>>>limitating to define "ring" as always containing a unity.
>>That is rather misleading because there are many known such constructions
>>with radically different properties and no natural way to prefer any one
>>of them in general.
>The construction Tonio had in mind -- and described at
>is indeed the universal one in the categorical sense.
This is known as the Dorroh extension. The point of my prior remark
is that this extension is often not of any use in many circumstances
since e.g. often it doesn't preserve crucual properties of the source
ring and, further, the extension is not minimal in various senses.
Such detriments are often discussed in the literature. Here is just
one of many such discussions:
W.D. Burgess; P.N. Stewart.
The characteristic ring and the "best" way to adjoin a one.
J. Austral. Math. Soc. 47 (1989) 483-496
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